where the trace operator on \(\partial \Omega \)  is \(\operatorname{tr}v = v|_{\partial \Omega }\)  for 0-forms, \(\operatorname{tr}v = (v \times \mathbf {n})|_{\partial \Omega }\)  for 1-forms, \(\operatorname{tr}v = (v \cdot \mathbf {n})|_{\partial \Omega }\)  for 2-forms, and \(\operatorname{tr}v = 0\)  for
3-forms [1]}, where \(\mathbf {n}\)  denotes the outward-facing unit normal
on \(\partial \Omega \) . In FEEC notation, the discrete spaces on the bottom row of
(REF ) are denoted as \(\mathrm {Q}^-_p \Lambda ^k(\mathcal {T}_h)\) . We denote the
discrete spaces employed as \(V^k_{h,p} \mathrm {Q}^-_p \Lambda ^k(\mathcal {T}_h) \cap V^k\) .
